#139113

order	512
nilpotency class	4
isoclinism	isoclinic class of #139074
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1746 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

357 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,9,6,8,4,14)(3,10,13,15,12,16,11,7)(17,18,23,30,26,35,22,34,25,31,21,28,24,19,32,29)(20,36,39,27,43,45,38,33,41,48,37,47,40,44,42,46);(1,3)(2,7,8,15)(4,11)(5,13)(6,12)(9,10,14,16)(17,19,25,35)(18,26,31,24)(20,33,41,46)(21,30,23,28)(22,34,32,29)(27,40,47,43)(36,42,48,38)(37,44,39,45);(2,8)(4,5)(7,15)(11,13)(17,20,25,41)(18,27,31,47)(19,33,35,46)(21,38,23,42)(22,37,32,39)(24,43,26,40)(28,48,30,36)(29,45,34,44)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=i,d^2=ghi,e^2=h,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,e^b=egh,f^b=fh,h^b=hi,d^c=dgh,e^c=ehi,f^c=fgh,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46309);